Problem: Solve for $x$ : $ 7|x + 9| - 8 = 2|x + 9| + 7 $
Subtract $ {2|x + 9|} $ from both sides: $ \begin{eqnarray} 7|x + 9| - 8 &=& 2|x + 9| + 7 \\ \\ { - 2|x + 9|} && { - 2|x + 9|} \\ \\ 5|x + 9| - 8 &=& 7 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 5|x + 9| - 8 &=& 7 \\ \\ { + 8} &=& { + 8} \\ \\ 5|x + 9| &=& 15 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x + 9|} {{5}} = \dfrac{15} {{5}} $ Simplify: $ |x + 9| = 3$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -3 $ or $ x + 9 = 3 $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -3 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -3 \\ \\ {- 9} && {- 9} \\ \\ x &=& -3 - 9 \end{eqnarray} $ $ x = -12 $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = 3 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& 3 \\ \\ {- 9} && {- 9} \\ \\ x &=& 3 - 9 \end{eqnarray} $ $ x = -6 $ Thus, the correct answer is $x = -12 $ or $x = -6 $.